How can I measure the noise on my AC mains with a 2ch oscilloscope?

OP: 

OP stands for "Original Poster" or the person who started this thread.

It's a good idea to tag those you reply to so they are aware. 

Shame on everyone discouraging you from experimenting.  Shame!  We need more hobbyist and tinkerers, not fewer, and we need more of them to have first hand knowledge of home power noise.  It also doesn't matter WHY the OP wants to tinker.  I remember many years ago I was thinking about making my own power regenerator.  It was a fun project and I would have learned a lot.  I went into an EE forum asking for help and one member immediately pulled out some thread I had typed in an audio forum to basically say no one should help me. 

Here we are now in an audiphile forum and I can't believe the number of opinions discouraging the OP.  I'm really disappointed in those of you doing so. Maybe this is just a learning experiment for the OP, and that's all it has to be.

Here is an article on doing so with a scope and an interesting thread from Stack Exchange and another from DIYaudio.

Hey OP!

I’ve done this before and I have some thoughts. DIYaudio is a better place for this type of hackery. You should have a 10:1 or 100:1 probe for your scope.

The thing I learned though is that an oscilloscope by itself is not actually very useful except in seeing really gross issues. For instance, I had an LED lamp which caused audible buzzing and you could see the "bite" the power supply took out of every positive going AC cycle. The bad news is that unless things are gross you don’t really get any sense of how good or bad the AC waveform is. For this you need to know things like harmonic distortion and/or a spectrum analyzer. First will give you a basic readout of how close to ideal the wave shape is, second will tell you more or less where in the frequency spectrum your problems are.

You might want to consider a PC based software solution, which uses the audio inputs, and maybe a custom, isolated interface to prevent the risk of injecting raw AC into your PC’s audio input. There are some good online resources for how to make one of these.  The limiting factor here is maximum frequency response will be limited to about 20 kHz. You won't see any RFI noise this way.